In a medium saucepan, combine scallion whites, ginger, garlic, soy sauce, rice vinegar, and 8 cups water; bring to a boil. Cover and reduce to a simmer. Cook until broth is flavorful, about 10 minutes.
Meanwhile, heat a grill pan over medium-high; lightly oil. Slice tofu in half; season with salt and pepper. Add tofu pieces to hot pan and cook, turning to grill all sides, about 15 minutes total. Remove from pan; once cool enough to handle, cut into small cubes.
Strain aromatics from broth with a slotted spoon; discard solids. Bring broth to a boil; add soy sauce and soba. Cook soba according to package directions. About 1 minute before the end of cooking, add bok choy, snow peas, and chile. Cook until veggies are tender-crisp. Add more soy sauce, if desired. To serve, place noodles in bowl with broth; top with tofu, scallion greens, and sesame seeds.